KMO 2015 Problem 5

For a fixed positive integer $k$ , there are two sequences $A_n$ and $B_n$ . They are defined inductively, by the following recurrences. $A_1 = k$ , $A_2 = k$ , $A_{n+2} = A_{n}A_{n+1}$ $B_1 = 1$ , $B_2 = k$ , $B_{n+2} = \frac{B^3_{n+1}+1}{B_{n}}$ Prove that for all positive integers $n$ , $A_{2n}B_{n+3}$ is an integer.